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Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. (English) Zbl 0981.93080
The paper is concerned with an optimal stochastic LQ control problem in infinite time horizon. In contrast to the deterministic case, the control and state weighting matrices in the cost functional are indefinite. This leads to an indefinite LQ problem which may be well-posed due to the nature of uncertainty involved. The problem gives rise to a stochastic algebraic Riccati equation (SARE) which is different from the classical algebraic Riccati equation. To analyze the SARE, some linear matrix inequalities (LMI’s) are introduced whose feasibility is shown to be equivalent to the solvability of the SARE. A computational approach to the SARE is developed via a semidefinite programming associated with the LMI’s. Finally, numerical experiments are reported to illustrate the proposed approach.
MSC:
93E20Optimal stochastic control (systems)
49N10Linear-quadratic optimal control problems
93B40Computational methods in systems theory
15A39Linear inequalities of matrices
90C22Semidefinite programming